The field of quantum computing is at an exciting time where we are constructing novel hardware, evaluating algorithms, and finding out what works best. As qubit technology grows and matures, we need to be ready to design and program larger quantum computer systems. An important aspect of systems design is layered abstractions to reduce complexity and guide intuition. Classical computer systems have built up many abstractions over their history including the layers of the hardware stack and programming abstractions like loops. Researchers initially ported these abstractions with little modification when designing quantum computer systems and only in recent years have some of those abstractions been broken in the name of optimization and efficiency. We argue that new or quantum-tailored abstractions are needed to get the most benefit out of quantum computer systems. We keep the benefits gained through breaking old abstraction by finding abstractions aligned with quantum physics and the technology. This dissertation is supported by three examples of abstractions that could become a core part of how we design and program quantum computers: third-level logical state as scratch space, memory as a third spacial dimension for quantum data, and hierarchical program structure.